amao abiodun thesis

8/11/2019 Amao Abiodun Thesis

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MATHEMATICAL MODEL FOR DARCY FORCHHEIMER FLOW WITHAPPLICATIONS TO WELL PERFORMANCE ANALYSIS

By

ABIODUN MATTHEW AMAO, B.Sc.

A THESIS

IN

PETROLEUM ENGINEERING

Submitted to the Graduate Facultyof Texas Tech University in

Partial Fulfillment of

the Requirements forthe Degree of

MASTER OF SCIENCE

IN

PETROLEUM ENGINEERING

Approved

Akif IbragimovChairperson of the Committee

Shameem SiddiquiCo-Chair of the Committee

Eugenio Aulisa

Lloyd Heinze

Accepted

John BorrelliDean of the Graduate School

August, 2007


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Texas Tech University, Abiodun Matthew Amao, August 2007

ii

ACKNOWLEDGEMENTS

This research was conducted at Texas Tech University under the supervision of

Dr. Akif Ibragimov and Dr. Shameem Siddiqui. I like to express my sincere thanks to Dr.

Akif Ibragimov and Dr. Eugene Aulisa who introduced this concept to me and supported

me with the mathematical framework for the thesis. Dr. Shameem Siddiqui and Mr.

Joseph McInerney were very helpful with the laboratory and experimental aspect of the

thesis. My sincere gratitude goes to the Chair of the Petroleum Engineering Department,

Dr Lloyd Heinze for his leadership and administrative prowess.

I am indebted to all members of staff and colleagues who contributed in one wayor the other to the success of my academic pursuit at Texas Tech University.

I deeply appreciate the moral support of my family back in Nigeria, my uncle

John Oyedeji, Nengi Harry and all loved ones and friends back home.

I appreciate the friendship and support of friends and members of my church in

Lubbock, International Christian Fellowship.

Finally and most reverently, I thank the Lord for His mercy, grace and blessings

which are too numerous for words.


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iii

TABLE OF CONTENTS

ACKNOWLEDGEMENTS ii

ABSTRACT vi

LIST OF TABLES vii

LIST OF FIGURES ix

LIST OF ABBREVIATIONS xii

CHAPTER

I. INTRODUCTION AND BACKGROUND 1

1.1 Background 31.2 Porous Media and Equations of Flow 6

1.3 Darcys Law: Assumptions and Limitations 7

1.4 Non-Darcy Flow; Darcy-Forchheimer Flow Equation 9

1.5 Flow Regimes in Porous Media 12

1.6 Significance of Thesis and Organization 14

II LITERATURE REVIEW 17

2.1 Non-Darcy Flow in the Reservoir 19

2.2 Flow in Fractures 23

2.3 Completions, Gravel Packs and Perforations 25

2.4 Beta Factor , its Measurement and Correlations 26

2.5 Non-Darcy Flow Modeling 30


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iv

III PROBLEM STATEMENT 34

3.1 Importance of Accurate Reservoir Pressure Prediction 35

3.2 Limitations of Current Techniques 36

3.3 Laboratory Experiments on Non-Darcy flow in Cores 37

3.4. Problem Statement 42

IV SOLUTION STATEMENT 44

4.1 Proposed Solution 44

4.2 Derivation of the Mathematical Model 444.3 Description of the Simulator 47

4.4 Numerical Computation and Algorithm 48

4.5 Laboratory Measurement of Beta Factor 50

V. RESULTS OF NUMERICAL COMPUTATIONS 55

5.1 Horizontal Well in a Rectangular Reservoir 55

5.2 Centered Circular Well in a Rectangular Reservoir 69

5.3 Off-centered Circular Well in a Rectangular Reservoir 72

5.4 Centered Circular Well in a Square Reservoir 75

5.5 Off-centered Circular well in a Square Reservoir 78

5.6 Concentric Well in a Circular Reservoir 81


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v

VI ANALYSIS AND DISCUSSION OF RESULTS 84

6.1 Analysis and Discussion of Experimental Results 84

6.2 Analysis and Discussion of Computational Results 96

VII. CONCLUSIONS AND RECOMMENDATIONS 106

7.1 Conclusions 106

7.2 Recommendations 107

REFERENCES 108

APPENDICES 115

A. RESULTS OF LABORATORY MEASUREMENTS OF ABSOLUTE

PERMEABILITY 115

B. ALGORITHM FOR SELECTION OF THE RIGHT BETA FACTOR

CORRELATION 136

C. EXPERIMENTAL SET UP AND EQUIPMENT USED IN THE

LABORATORY 138

D. VITA 141


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vi

ABSTRACT

Well performance and productivity evaluation is a fundamental role of petroleum

engineers and this is done at different phases of petroleum production; from the reservoir

to the well bore through the tubulars and ultimately to the stock tank. This task requires

physical and mathematical models that adequately characterize oil and gas flow at these

different phases of petroleum production.

This thesis reviews different scenarios where the effects of non-linearity in flow

are apparent in petroleum and gas reservoirs and cannot be neglected any more.

Laboratory experiments were carried out on core samples to show non-linearity in flow,which confirms deviation from the traditional Darcy law, used in reservoir flow

modeling.

Historically non-Darcy flow has only been reckoned with in high flow rate gas

wells, in which it has been treated as a rate dependent skin factor and has been assumed

to act only in the vicinity of the well-bore, while neglecting the reservoir. This work

seeks to show the inherent errors due to the negligence of this phenomenon, which is

fundamental to the calculation of the productivity index of the well. Using the modified

non-linear Darcy law as the equation of motion to model filtration in porous media, this

new model is compared to the conventional Darcy law. The proposed method delivers

robust framework to model non-linear flow in the reservoir.

The result of this project will equip reservoir engineers with a robust technique to

analyze well performance; this approach will provide better evaluation tool for selecting

wells for remedial operations such as work-over or stimulation.


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vii

LIST OF TABLES

3.1 Result of non-Darcy flow experiment on core #13 37



4.1 Porosity, physical properties and Lithology of core samples used 52

4.2 Porosity ranking and cores used for permeability measurements 53

4.3 Porosity and permeability of cores samples used in beta factor experiment 54

5.1 Productivity index at different drain hole lengths 57

5.2 Productivity index @ L=5000cm at different rates and beta values 595.3 Productivity index @ L=10,000cm at different rates and beta values 61

5.4 Productivity index @ L=20,000cm at different rates and beta values 63



5.7 Productivity index at different rates and beta values for Geometry 5.2 70





6.1: Beta factor correlations used for analysis 84

6.2: Calculated beta values using the nine correlations 85

A.1: Experimental results of permeability measurement on core #1 115




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viii









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ix

LIST OF FIGURES

1.1 Flow regimes in porous media after Basak (1977) 14

3.1 Experimental result of non-linearity in flow through core #13 38

3.2 Experimental result of non-linearity in flow through core #9 40

3.3 Experimental result of non-linearity in flow though core #26 42

4.1 Flow chart of numerical computation 49

4.2 Experimental setup for permeability and beta factor experiments 50

4.3 Procedure for laboratory measurement of beta factor 51

5.1 Geometry of the horizontal drain in a rectangular reservoir 565.2 Plot of productivity index at different drain hole lengths 58

5.3 Productivity index vs. rate @ L=5000 cm 60





5.8 Circular well in a rectangular reservoir (Geometry 5.2) 69

5.9 Productivity index plot for Geometry 5.2 71

5.10 Off-centered circular well in a rectangular reservoir (Geometry 5.3) 72


5.12 Circular well in a square shaped reservoir (Geometry 5.4) 75


5.14 Off-centered circular well in a square reservoir (Geometry 5.5) 78



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5.16 Circular well in a circular reservoir (Geometry 5.6) 81


6.1: Calculated beta factors for core #10, using the correlations 86






6.7: Calculated beta factors for core #13, using the correlations 926.8: Calculated beta factors for core #23, using the correlations 93



6.11: Productivity Index versus length for different rates at =0 97

6.12: Productivity Index versus length for different rates at =2.4 98



6.15: Comparison of Productivity Index for all Geometries used at = 0 102

6.16: Comparison of Productivity Index for all Geometries used at = 2.4 103



A.1: Darcys law plot for core #1 116

A.2: Klinkenberg correction plot for core #1 117

x


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xi









A.11: Darcys law plot for core #13 126A.12: Klinkenberg correction plot for core #13 127









B.1: Beta Factor Correlation Selection Chart 137

C.1: Gas Permeameter, Hassler core holder and bubble flow tube 138

C.2: Helium Porosimeter 139

C.3: The core samples used for the experiments 140


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LIST OF ABBREVIATIONS

Symbol Definition

A Cross-sectional Area

Bo Oil formation volume factor

Bg Gas formation volume factor

d Average grain diameter

D Non-Darcy flow coefficient

F Flux

F ND Non-Darcy Fluxh height of fluid head

h Reservoir thickness

J Productivity index

K Permeability

L Length of Core/ Sand bed

M Gas Molecular weight

P Pressure

R P Average reservoir pressure

P wf Well flowing pressure

q Production rate

RE N Reynolds number

r d Reservoir drainage radius

r e External boundary radius

r w Well bore radius

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xiii

S Skin factor

St Total Skin

t Time

T Temperature

v Flow velocity

x, y, z Rectangular coordinates

Z Gas compressibility factor

Greek Letter

Fluid density

Alpha

Inertial factor

Viscosity

Porosity

Tortuosity


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xiv

Subscript

o Oil

g Gas

w Water

sc Standard conditions

f Fracture


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CHAPTER I

INTRODUCTION AND BACKGROUND

The analysis and prediction of reservoir and well performance requires diverse

information which a reservoir or a production engineer must have before he/she can

adequately analyze reservoir performance or predict future production under various

production mechanisms, the key to which is a consistent and representative mathematical

model of the physical parameters governing flow in the reservoir.

Several techniques by which reservoir parameters can be acquired have been

devised. These include core analysis, well logging and pressure transient testing/analysis;of these techniques, pressure transient analysis gives the most representative information

on the reservoir at a scale consistent with the size of the reservoir.

Pressure transient testing is simply generating and measuring pressure variation

with time in wells after a characteristic disturbance has been generated in the well;

analysis of the generated data leads to an estimation of rock, fluid, well and reservoir

properties which are required in well performance engineering.

Information obtained from transient testing include well-bore volume, skin,

damage/improvement, reservoir pressure, permeability, porosity, reserves, reservoir and

fluid discontinuities which are key input in reservoir performance analysis, well

improvement schemes, economic analysis and production forecast.

Historically, in oil field practice the productive capacity of producing wells is

generally evaluated using the productivity index (PI), defined as the rate of production

per unit pressure drop. It has the symbol J , and it is expressed mathematically as:


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wf R P P q

J = (1.1)

Where q = Production rate

R P = Average reservoir pressure

P wf = Well flowing pressure

And based on Darcy law, the productivity index is given by;

S r r

B

hk P P

q J

w

e

av

wf R +

=

=

43

ln2.141

(1.2)

Where,

k av = Average permeability

S = Skin factor

The productivity index J for different reservoir geometry, based on the shape

factor is given as;

+

==S

r C

A B

hk P P q

J

w A

av

wf

4306.10

ln21

0078.0

2 (1.3)

Where,

C A = Shape factor

A = Drainage area

The productivity index has been traditionally calculated based on the fundamental

assumption of the validity of Darcys law in porous media.

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However, Darcys law breaks down under conditions of high velocity flow which

is proven to exist in gas wells, high permeability reservoirs, fractured reservoirs

(naturally and hydraulically fractured) and in perforations, especially near the well bore.

This work seeks to review the dynamics of non-Darcy flow and how it affects the

productivity index calculation and well performance prediction in different reservoir

geometry and scenarios.

1.1 Background

The physics of fluid flow in different media and conduits is a well researched areain engineering with groundbreaking works by pioneer workers in this field of

engineering. Equations describing flows in media such as cylindrical pipes, rectangular

conduits, and other forms and shapes of conduits have been developed analytically over

the years.

The three fundamental principles governing flow in any media and upon which

the development of these flow equations are based are:

(a) Law of conservation of mass or the continuity equation

(b) Equation of state of the fluid

(c) Law governing the dynamics of fluid flow or Newtons law

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Mathematical expression and statement of these laws are given below:

(a) Law of conservation of mass or the continuity equation

This law states that the net excess of mass flux, per unit time into or out of any

infinitesimal volume element in the fluid system is exactly equal to the change per unit

time of the fluid density in that element multiplied by the free volume of the element,

stated mathematically as:

dt dz vd

dy

vd

dxvd

v z y x

=++= )()()().( (1.4)

(b) Equation of State

This is the equation that describes the fluid and its thermodynamic flow properties

as it relates to pressure, temperature and density. It is stated simply as;

0),,( =T P f (1.5)

(c) Law governing the dynamics of fluid flow (Newtons Law)

This law imposes on the velocity distribution in every flow system the

requirement of a dynamical equilibrium between the inertial forces and the viscous forces

and those due to external body forces and the internal distribution of fluid pressures. This

law takes into account all the forces acting on the fluid as it flows in the medium, the

forces acting on an elemental fluid particle and their equations are;

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(i) Pressure gradients in the coordinates of flow

dz dp

dydp

dxdp

,,

(ii) External body forces, such as gravity in the direction of flow

z y x F F F ,,

(iii) Forces opposing motion or viscous forces, due to internal resistance of the

fluid to flow. An expression for viscous flow is given by:

dxd

v x

312 + ,

dyd

v y

312 + ,

dz d

v z

312 +

where,

2

2

2

2

2

22

dz d

dyd

dxd ++ and

dz dv

dy

dv

dxdv

v z y x ++== . (from the continuity equation)

The flow equation is obtained by equating the sum of these three forces stated

above to the product of mass and acceleration of the volume element of the fluid,

therefore for an elemental fluid particle, the acceleration is given by the total time

derivative of the velocity given by,

dz d

vdyd

vdxd

vdt d

dz d

dt dz

dyd

dt dy

dxd

dt dx

dt d

Dt D

z y x +++=+++

Combining these parameters gives the Navier Stokes equation in three dimensions

dxd

v F dxdp

Dt Dv

x x x

312 +++= (1.6a)

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dyd

v F dydp

Dt

Dv y y

y

312 +++= (1.6b)

dz d

v F dz dp

Dt Dv

z z z

312

+++= (1.6c)

The three laws and equations stated above are mathematically and scientifically

sufficient to predict all the parameters of the flow of a viscous fluid flowing through a

medium of any shape, size or geometry.

The particular solution of the partial differential equations stated above for a

given medium is only possible when the boundaries of such a medium are clearly

defined. That is, the fluid system and the detailed physical conditions that serve as the

initial conditions of the system must be known before a solution can be obtained for any

flow medium or geometry.

1.2 Porous Media and Equations of Flow

A porous medium can be defined as a solid body which contains void spaces or

pores that are distributed randomly; without any conceivable pattern throughout the

structure of the solid body. Extremely small voids are called molecular interstices and

very large ones are called caverns or vugs. Pores (intergranular and intercrystalline) are

intermediate between caverns and molecular interstices.

Fluid flow can only take place in the inter-connected pore space of the porous

media; this is called the effective pore space.

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Petroleum reservoirs are porous media and the storage and flow of hydrocarbons

takes place in these pore spaces which serve as conduit to the flow of oil, gas and water

during production or the depletion of a reservoir. Some peculiarities of the porous

media encountered in petroleum reservoirs are:

(a) There is no geometry or geometrical quantity that can characterize or describe

the system of pores in any porous body.

(b) The pore walls are always irregularly converging or diverging and are highly

irregular in any cross-section.

(c) Visualizing pores as cylindrical tubes is not consistent with any pore systemknown in nature.

These inherent and attendant characteristics of a porous medium makes it grossly

impossible to solve the system of partial differential equations (1.4 ), (1.5) and (1.6)

describing the general fluid flow phenomena stated earlier.

Literature is replete with several simplifying assumptions made by earlier

researchers to relate the pores in porous media to known shapes or geometry for which

analytical or numerical solution has been gotten, but none of these rightly solves the

porous media problem.

1.3 Darcys Law: Assumptions and Limitations

Henri Darcy, a French civil engineer, in his 1856 publication laid the real

foundation of the quantitative theory of the flow of homogenous fluids through porous

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media. As a civil engineer, he was interested in the flow characteristics of sand filters

used to filter public water in the city of Dijon in France.

The result of his classic experiments, globally known as Darcys law, is thus

stated: The rate of flow Q of water through the filter bed is directly proportional to the

area A of the sand and to the difference h in the height between the fluid heads at the

inlet and outlet of the bed, and inversely proportional to the thickness L of the bed.

This can be stated mathematically as:

LhCA

Q = (1.7)

where C is a property characteristic of the sand or porous media.

Darcys law represents a linear relationship between the flow rate Q and the head

(pressure gradient) L

h.

The constant of proportionality C in the original Darcy equation has been

expressed as k

, where is the viscosity of the fluid and is called the permeability

of the porous medium. Permeability is a property of the structure of the porous media

and it is entirely independent of the nature of the fluid. It uniquely sums up the

geometric properties of the porous media such as porosity, shape of the grains, size of

the grains and the degree of cementation. The permeability k is considered to

completely and uniquely characterize the dynamic properties of a porous media with

respect to flow of fluids though it.

k

Hence, Darcys law is stated as:

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dl dpk

v

= (1.8)

And more generally as:

dxdpkA

q

= (1.9)

Darcys empirical equation is a statistical average of classical hydrodynamic

equation over the minute and detailed variation occurring in the individual pores; it

gives a simplified macroscopic representation.

Inherent in the development of the Darcy flow model are the following assumptions;

a) Darcys law assumes laminar or viscous flow (creep velocity); it does not

involve the inertia term (the fluid density). This implies that the inertia or

acceleration forces in the fluid are being neglected when compared to the

classical Navier-Stokes equations.

b) Darcys law assumes that in a porous medium a large surface area is exposed

to fluid flow, hence the viscous resistance will greatly exceed acceleration

forces in the fluid unless turbulence sets in.

1.4 Non-Darcy Flow; Darcy-Forchheimer Flow Equation

Darcys empirical flow model represents a simple linear relationship between

flow rate and pressure drop in a porous media; any deviation from the Darcy flow

scenario is termed non-Darcy flow.

Physical causes for these deviations are grouped under the following headings 31;

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a) High velocity flow effects.

b) Molecular effects.

c) Ionic effects.

d) Non-Newtonian fluids phenomena.

However, in petroleum engineering, the most common phenomenon is the high

flow rate effect. High flow rate beyond the assumed laminar flow regime can occur in the

following scenarios in petroleum reservoirs.

a) Near the well bore (Perforations)

b) Hydraulically fractured wellsc) Gas reservoirs

d) Condensates reservoirs (Low viscosity crude reservoirs)

e) High flow potential wells

f) Naturally fractured reservoirs

g) Gravel packs

It is therefore imperative for reservoir engineers to develop a better flow model

that is adequately representative and uniquely characterizes the physical parameters and

variables in these flow scenarios.

In 1901, Philippe Forchheimer, a Dutch man, while flowing gas thorough coal

beds discovered that the relationship between flow rate and potential gradient is non-

linear at sufficiently high velocity, and that this non-linearity increases with flow rate. He

initially attributed this non-linear increase to turbulence in the fluid flow (it is now known

that this non-linearity is due to inertial effects in the porous media), which he determined

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to be proportional to , with a being a constant of proportionality. Cornel and Katz 6

gave a value of to a, where (beta) is called the inertial factor and is the density

of the fluid flowing through the medium.

2av

The additional pressure drop due to inertial losses is primarily due to the

acceleration and deceleration effects of the fluid as it travels through the tortuous flow

path of the porous media. The total pressure drop is thus given by Forchheimer empirical

flow model stated traditionally as;

2vvk dx

dp += (1.11)

This can also be written in vector notation as:

P vvv =+ rrr (1.12)

Wherek

= ,

The Forchheimer equation assumes that Darcys law is still valid, but that an

additional term must be added to account for the increased pressure drop. Hence this

equation will be called the Darcy-Forchheimer flow model in this thesis.

Equation (1.11) is based on fitting an empirical equation through experimental data.

However, Forchheimer based on these data set later propose a third order equation

given by:

32 cvbvavdx

dp ++= (1.13)

where a, b and c are constants as in equations (1.11) and (1.13) above.

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Another flow model that has been proposed for flow in porous media is the power

law model, given by:

navdx

dp = (1.15)

where n has a value between 1 and 2

In vector notation, it is stated as:

P vvC nn = r1 (1.16)

However, of these three models the most widely used is given by equation (1.11)

and it will form the basis of analysis in this project to characterize high velocity non-

Darcy flows in porous media.

1.5 Flow Regimes in Porous Media

Analogous to flow in pipes and conduits, several researchers have also tried to

define a flow regime in porous media to distinguish flow regimes and to predict the onset

of one or the termination of another. Typically for flow in pipes and conduits, the

Reynolds number is used to delineate flow regimes. A Reynolds number less than 2100

implies laminar flow, while a greater number implies turbulent flow. In porous media

however, there is no clear limit or a magic number that defines this transition. The non-

linearity experienced in non-Darcy flow is not a result of turbulence but inertia effects as

stated earlier, hence non-Darcy flow is known to occur in porous media at a much more

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lower Reynolds number, and it is not initiated by a change in flow regime. The Reynolds

number in porous media is given by;

vd N =Re (1.17)

where d is average grain diameter of the grains in the porous media. However for a media

with non-Darcy flow (e.g. a fracture) the Reynolds number is given by;

k v

N =Re (1.18)

This is just another Reynolds number with the characteristic length defined by k .

In the literature, depending on the flow velocity and the nature of the porous

media different flow patterns have been observed. However four major regimes were

proposed by Dybbs and Edwards (using laser anemometry and visualization technique).

These four regimes are;

a) Darcy or laminar flow where the flow is dominated by viscous forces, here the

pressure gradient varies strictly linearly with the flow velocity. The Reynolds

number at this point is less than 1.

b) At increasing Reynolds number, a transition zone is observed leading to flow

dominated by inertia effects. This begins in the range Re=1~10. This laminar

inertia flow dominated region persists up to and Re of ~150.

c) An unsteady laminar flow regime for Re =150 ~ 300 is characterized by

occurrence of wake oscillations and development of vortices in the flow profile.

d) A highly unsteady and chaotic flow regime for Re > 300, it resembles turbulent

flow in pipes and is dominated by eddies and high head losses.

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However there is large variation in the limiting Reynolds number for these

transition zones as published in the literature, therefore one cannot be too categorical

about limits and transition zones as it relates to the Reynolds number in porous media.

Figure 1.1 below is a diagrammatic representation of the flow regimes in a porous

media as proposed by Basak 49.

Post-Darcy ZonePre-Darcy Zone Darcy Zone

Forchheimer

Laminar

Pre-Laminar

Turbulent

No Flow

Figure 1.1: Flow Regimes in Porous Media after Basak (1977)

1.6 Significance of Thesis and Organization

The results and knowledge gained from this thesis will be useful in adequately

evaluating production performance of wells and aid reservoir engineers in modeling

reservoir flow with more robust equations. Selection of candidate wells for well

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engineering routines will be more objective and representative of actual scenario in the

reservoir. The findings from this thesis will further illuminate known discrepancies in

well test analysis and help to ratify a fundamental source of uncertainty in well test

models.

This thesis is organized into seven chapters; the contents of each chapter are

summarized.

Introduction and background; this chapter contains a brief introduction to the

fundamental principles of fluid flow in porous media, with a review of governing

equations of flow in porous media as it relates to Darcy and non-Darcy flows.Literature review; this is an assessment of current industry practice and

methodology used to handle non-Darcy flow in different scenarios in the petroleum

industry with a review of non-Darcy flow modeling in the literature.

Problem statement; a categorical expression of the problem this thesis seeks to

solve, with the motivation and importance of this solution to the petroleum industry.

Solution statement; this is a procedural statement of the development of a

proposed solution to the stated problem and why this approach is significantly different

from previous approaches. It also gives a statement of the results expected using this

procedure.

Results; a catalogue of results obtained during laboratory experiment on core

samples and numerical simulations of various reservoirs and well geometries.

Discussion and analysis of results; the results obtained are compared with current

industry practices and discussed.

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Conclusions; the final chapter summarizes the thesis and presents the conclusions

drawn.

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CHAPTER II

LITERATURE REVIEW

In the early days of the petroleum industry it was noted that the pressure drop

measured in the vicinity of the wellbore was greater than the pressure drop computed

using industry-wide modeling equations 36. This excessive pressure drop was explained by

assuming a decrease in permeability (formation alteration) due to formation damage in

the vicinity of the wellbore. The capacity of a well to produce is generally accepted to be

directly proportional to the pressure drop in the reservoir. Hurst and Van Everdingen 36 in

the 1950s introduced a dimensionless term called the skin factor which was used toexplain this phenomenon 36. The skin factor (S) was originally designed to give a

numerical value to the additional resistance assumed to be concentrated around the

wellbore resulting from drilling and completion techniques employed or the production

practices used. This ultimately leads to an additional pressure drop, this pressure drop is

called the skin effect. The magnitude of the skin effect determines the productive

capacity of a well. This has also been used in well performance evaluation and remedial

operations.

Over the years, the skin factor has been broken down into several components. An

expression for the total skin (S) is given below:

S = S c + S p + S d + S G + S A+ So (2.1)

Where,

S= skin

Sc= completion skin due to partial penetration

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S p= perforation skin

Sd= skin due to damage around the well bore

SG= gravel-pack skin

SA= outer boundary geometry skin

So= slanted well skin

The additional pressure drop due to high velocity flow is also expressed as an

equivalent skin, Dq; where q is the flow rate and D is a composite of the following high

velocity flow terms;

D = D R + D d + D dp + D G (2.2)Where

DR = reservoir high velocity flow term beyond the well bore area

Dd= damaged zone high velocity flow term

Ddp= high velocity flow term in the region surrounding the perforations

DG= high velocity flow term in a gravel packed perforation

q = flow rate

Assuming all the other skin sources are summed up in S, therefore, for the case of high

velocity flows, the total skin factor will be given by;

St = S + Dq (2.3)

Where;

St = Total skin

Dq = rate dependent skin factor

D = Non-Darcy flow coefficient

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It is obvious that the value of the rate dependent skin (Dq) will not be a constant,

in comparison to the mechanical skin, as it will depend on the flow rate, in a direct

proportionality. This will subsequently vary the value of the total skin S t.

As can be seen from the sources of skin enumerated above, the petroleum industry

has known the inadequacy of Darcys law to adequately predict the pressure loss at high

flow rate; however, this skin factor has been assumed to be concentrated in the vicinity of

the wellbore i.e. at the sandface or across the completion, the effect of non-Darcy flow in

the reservoir has been neglected and assumed to be negligible.

The treatments of non-Darcy flows will be reviewed under the scenario wherethese effects come into play in reservoir engineering.

2.1 Non-Darcy Flow in the Reservoir

Non-Darcy flow occurs in petroleum reservoirs that have high conductivity to

flow. Initially it was assumed that this phenomenon was only relevant to gas wells, but

field observations and analysis show that it relevant to oil wells as well. This was proven

by Fetkovitch during a comprehensive field study of 40 oil wells 10.

As narrated above, non-Darcy flow has been treated as a rate dependent skin

factor by the inclusion of the term Dq as an additional source of pressure loss in the

vicinity of the wellbore. The various techniques for evaluating this parameter are

reviewed below.

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2.1.1 Multi-rate Tests

Multi-rate tests are traditionally used to evaluate the deliverability of a gas or oil

well, the additional pressure drop due to non-Darcy effect is calculated from the

Houpeurt (back-pressure) analytical equation and from the empirical equation proposed

by Rawlins and Schellhardt in 1936. These tests are listed below;

(i) Flow after flow test

(ii) Isochronal test

(iii) Modified isochronal test

2.1.1.1 Flow-After-Flow Tests

This is also called the gas back pressure of four point test, it is conducted by

producing the well at a series of different stabilized (pseudo-steady state) flow rates and

measuring the stabilized bottom hole flowing pressure at the sand face. Each flow rate is

established in succession, often conducted with a sequence of increasing flow rates. A

major limitation of the test procedure is that the well must reach a stabilization period,

especially in low-permeability formations that take longer to reach stabilization.

Schellhardt and Rawlins of the USBM developed an empirical equation for analyzing

back-pressure data based on field data analysis. They proposed a relationship which

applicable only at low pressures is given by

n s f P P C q )(

22

= (2.4)

Where,

C= Stabilized performance coefficient

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n = inverse slope of log-log plot of ( ) versus q22 s f P P

The theoretical value of n ranges from 0.5, which indicates non-Darcy flow regime, to 1.0

indicating a flow regime governed by Darcys law

A much more consistent analytical equation developed from the gas diffusivity

equation was proposed by Houpeurt which is stated as;

222 g g wf R Bq Aq P P += (Gas wells) (2.5)

222 R P oowf Bq Aq P += (Oil wells) (2.6)

Where,

h K x B

S r r

Ao

ot

w

e3

0

1008.775.0ln

+

=

Dhk x

B B

o

oo31008.7

=

+

= t

w

e

g

g S r r

hk x

T z A 75.0ln

1003.7 4

Dhk x

T z B

g

g

41003.7 =

q

P P wf R22

A Cartesian plot of ( ) against q gives a plot with intercept A and slope B,

from which the value of D, can be calculated knowing all other variables.

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2.1.1.2 Isochronal Tests

s proposed by Jones, Blount and Glaze 19. This test was

is long

me pe

.1.1.3 Modified Isochronal Tests

This technique wa

designed to shorten the stabilization time required for the flow after a flow test. Th

time is usually impractical in some cases, especially in low-permeability reservoirs. It is

conducted by alternating producing the well, then shutting the well in and allowing it to

buildup to the average reservoir pressure before the beginning of the next flow period.

Pressures are measured at several time increments during each flow period. The

ti riod in which the pressures are monitored is the same relative to the stating time

of each flow period. The same method of analysis is used to analyze the data to obtainvalues for D.

2

d in a paper by Brar and Aziz. It is a modification of

e isoc

ds

is known to be less accurate than the isochronal test, due to this short time

st

This technique was propose

th hronal test aimed at shortening the test times required for the well to build up to

the average reservoir pressure in the drainage area of the well. It is conducted like an

isochronal test, except that the shut in periods are of equal duration and the flow perio

are of equal duration. The length of the shut-in period usually equals or exceeds the flow

periods.

It

periods allowed for pressure build up. The data analysis is the same as the previous te

types.

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2.1.2 Single Well Test Techniques

The use of a single we factor has been

34,

2.1.3 Correlations

Ramey proposed an equation fo -Darcy flow coefficient if

ulti-te

ith

ll test to estimate the non-Darcy skin

proposed by several researchers. These include Camacho et al, Warren, Spivey et al

Kim and Kang 21. They proposed new methods for using single well tests to obtain the

rate dependent skin factor, based on the algorithms they developed.

r calculating the non

m st data are not available. The expression was obtained by integrating theForchheimer equation for the drainage radius r d to the well bore r w. However, he

confirmed that the result may be in error of about 100%, based on a comparison w

multi-rate tests. The expression is given as;

w sc

sc k Mp x Dhr T

1510715.2 = (2.7)

where the variables have the usual notations.

2.2 Flow in Fractures

The occurrence of non-Da n fractures is well documented

tural or

rcy flow phenomenon i

in the literature. Early workers have come to understand the importance of this

phenomenon as it affects the productivity of fractures. Fractures can either be na

induced e.g. hydraulic fractures. The two distinct flow regimes observed during well tests

in fractured reservoirs point to the fact that the flow regime in the matrix is different from

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the flow regime in the fracture, although this has been thought to affect only the high rate

wells.

Hydraulic fracturing is a widely used completion method in the tight gas

rmati

on-

2.2.1 Hydraulic FracturesIn hydraulic fracture stim productive capability and

ells.

ure

.

fo ons all over the world. Several hydraulic fracturing jobs are implemented

annually. However, the performances of these fractures are highly dependent on n

Darcy flow effects in the fracture. Several ongoing studies are looking into how to

maximize fracture design and mitigate the non-Darcy effect in fractures.

ulation of wells, the wells

overall reserve recovery is impacted by non-Darcy flow as it causes a reduction in the

propped half length to a lower effective half length. Fracture design engineers have

historically neglected this phenomenon assuming that it only impacts high velocity w

According to Vincent et al. 37, ignoring the non-Darcy effects while designing

fractures will lead to inaccurate production forecasts, suboptimal fracture design and

selection of inappropriate proppant type. They opined that fluid velocities in real fract

are approximately 1000 times greater than laboratory measurements; hence laboratory-

measured proppant permeability values are not really suitable when designing fractures

Miskimins et al. 26 in their investigation of flow rates at which non-Darcy flow

influences retained fracture permeability discovered that its effect is significant across a

wide spectrum of flow rates from as low as 50-100 MCFD, and these decrease can range

from 5% at a flow rate of 50 MCFD to 30% at 400 MCFD under a given set of

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conditions. Presently in fracture design non-Darcy flow is integrated by accurate

selection of a proppant type based on laboratory tests and field observation with

particular emphasis on the beta factor of the proppant to be used.

To optimize fracture design, Lopez-Hernandez et al. 24 prop

osed a beta factor

ethod by;m to calculate the effective fracture permeability eff f k . This parameter is given

g

g f

f eff f vk

k k

+=

1 (2.8)

This expression was derived by combining the Darcy and non-Darcy flow equations in a

eability when

of

2.3 Completions, Gravel Packs and Perforations

fracture and solving for eff f k , which determines the actual pressure drop in the fracture.

Another fracture gn criterion is to minimize the pressure loss due to the inertia

losses by minimizing the 2v term in the traditional Darcy-Forchheimer equation. This

can be achieved by selecting a proppant with an optimal beta factor.

The beta factor may be more important than the reference perm

desi

selecting proppant for a fracturing job. Hence it is imperative to know the beta factor

the proppant to be used in the design, as they are not usually reported in the industry.

Several work ions and

occur

l

ers have investigated non-Darcy flows in complet

perforations. It was observed that large pressure drops in perforated completions

mostly in the convergence zones and the in perforation tunnel, especially in high rate oi

and gas wells. Nguyen 29 experimentally studied non-Darcy flow in perforations. He

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discovered that non-Darcy flow in perforations is a function of perforation geometry,

permeability of the gravel. In his experiments, he used water and air as the flowing fluid

and came to the conclusion that the relationship between pressure drop and flow rate is

non-linear. Therefore, a simplistic analysis of the flow using Darcys law will over

predict the productivity and cases have been found where the productivity has been

predicted by as much as 100%.

In well performance engine

and

over-

ering of gravel packed completions, it is important to

2.4 Beta Factor , its Measurement and Correlations

delineate the pressure drop due to mechanical skin or rate dependent skin (non-Darcy

flow) so that the right remedial action can be taken to improve the productivity of thewell.

The beta fa ditional Darcy-

cient.

d

ression for the beta factor falls under two broad categories;

empirical and theoretical models. The theoretical models are further divided into parallel

and serial models.

ctor , which is a constant of proportionality in the tra

Forchheimer equation, was first proposed by Cornel and Katz 6. It is known by several

names which include; non-Darcy flow coefficient, inertial flow coefficient and the

turbulence factor. However, in these thesis we will adopt the non-Darcy flow coeffi

It is widely agreed that is a property of the porous media; it is a strong function of the

tortuosity of the flow path and it is usually determined from laboratory measurements an

multi-rate well tests.

The derived exp

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In the parallel model, the porous medium is assumed to be made up of straig

capillary bundles of uniform diameter. According to Li and Engler 22, based on the work

of Ergun et al., and

ht

Polubarinova-kochina, an expression for the Beta factor for a parallel

model is given by;

5.15.0

K c= (2.9

Where c is a constan

)

t

In the serial type model, the pore space is serially lined up; capillaries of different

pore types are aligned in series. Li et al. also proposed an expression for the Beta factor

d on the work of Scheidegger, the beta factor is given as;

22

for a series model base

There are several empirical correlations in the literature used to predict the beta

factor. These expressions differ due to the varied experimental procedure, porous media

stently shown that permeability,

riments on 355 sandstone and 29 limestone cores (vuggy,

rystalline, fine grained sandstone) and came up with a correlation given by

K c ''= (2.10)

Where ''c is a constant related to pore size distribution

and fluids used for the experiments. However, it is consi

porosity and tortuosity are the main parameters on which the beta factor depends. Also,

some correlations have been developed for multiphase flows, hence these correlations are

function of saturation as well.

2.4.1. Permeability Defined Beta Factors

Jones 19 conducted expe

c

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55.1

101015.6

K x= (2.11)

Pascal et al, based on mathematical analysis of data from Multirate wells in

Where K is in md and in 1/ft

hydraulically fractured reservoirs, proposed a correlation given by,

176.1

12108.4 K

x= (2.12)

Cooke based on his experiments in using brines, reservoir oils and gases in

(2.13)

ments,roposed the correlation given as,

(2.14)

l porous media proposed to use the following

tions:

(2.15)

Where K is in md and is in 1/cm.

Where K is in md and is in 1/m.

propped fractures, predicted the non-Darcy coefficient as,

abK =

Where a and b are constants determined by experiments based on proppant type.

2.4.2. Correlations Based on Permeability and Porosity

Eguns empirical equation based on data found in the literature and experi p

2/1 10(= ab 2/32/18 ) K

Where a=1.75, b=150, K in Darcy and in 1/cm.

Janicek and Katz, for natura

equa

4/34/581082.1 = K x

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Geertsma based on his experiments on consolidated and unconsolidated

sandstones, dolomites and limestone and a review of other works, he proposed an

pirical correlation giveem n by:

5.55.0

005.0 =

K (2.16)

where K is in cm 2 and in cm -1

2.4.3. Correlations Based on Permeability, Porosity and Tortuosity

Liu et al further worked on the data

onsidering the effect of tortuosity they got a better

orrelation given as,

used by Geertsma, Cornell and Katz, Evans

and Evans and Whitey, and by c

c

x 8101

K

9.8= (2.17)

Where is in ft -1 and K in md

they proposed a correlation given by,Others include, Thauvin et al.,

29.098.0

35.341055.1

K x= (2.18)

-1

the literature. In choosing a correlation to use in predicting the non-Darcy coefficient, Li

.22 proposed the fo

of the formation (e.g. from well logs)

Where is in cm and K in Darcy

This is not an exhaustive listing, there are several other correlations proposed in

et al llowing guidelines.(see Appendix B)

(a) Determine the lithology

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(b) Determine what parameters are known or can be found, use the correlation

that has as many known parameters as possible.

(c) Determine the pore geometry of the formation and the relativity of flow

ling

direction to pore channels.

2.5 Non-Darcy Flow Mode

e

Darcy f y equation has been widely used in well test models,

servoir simulation models and all other petroleum engineering models to simulate fluidflow in the reservoir. One im predict reservoir pressure

is

er

rchheimer equation for non-Darcy

Fluid flow in porous media in the petroleum industry has been modeled by th

low equation. The diffusivit

re portant use of these models is to

and other reservoir parameters that are required for well performance evaluation and

prediction. Muskat 27 was the first to utilize Darcys law in deriving fluid flow equations

in oil and gas reservoirs for different flow patterns and reservoir geometries. This has

served the petroleum industry for a long while. However recent research and further

insight into non-Darcy flow phenomenon in the reservoir and scenario where it occurs

necessitating a new look into this historical trend.

Numerical modeling of non-Darcy flows began in the 1960s; some of the pione

workers include Smith, Swift et al., who investigated the effects of gas flow on well

testing. Researchers in recent times are looking at newer and better ways of modeling

fluid flow in porous media while integrating the Fo

flow. Thus they are developing a new diffusivity equation that can be used in reservoir

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simulators and other numerical models so that more accurate and better predictive mod

can be obtained.

Belhaj et al. 5 developed a new diffusivity equation that was used to model non-

Darcy flow in the reservoir. They used a finite difference modeling scheme, based on the

Crank-Nicholson

els

and Barakat-Clark numerical modeling methods, while comparing both

Darcy and non-Darcy flows. They derived a new expression for the diffusivity equation

based on the Darcy-Forchheimer equation in two dimensions stated as;

y P

x P

t P

K y P

x P 22

Based on the results of their numerical simulations, they opined that the

++ +=+ vvc 222

Forchheimer

model gave more realistic result for all ranges of pressure gradients, flow rates,

permeabilities, porosities, viscosity and fluid density.

g

ling. He

voir and also at the well bore

Su33 of Saudi Aramco, in his publication detailed how non-Darcy flow modelin

can be integrated into a reservoir simulator, especially for multiphase flow mode

modeled both the rate dependent skin factor in the reser

treating the two differently. He took the non-Darcy consideration into account, both in

the cell to cell flux and in the vicinity of the well bore. His model also proposed the

Darcy-Forchheimer equation for each phase flowing in the reservoir; his phase based

non-Darcy flow equation is given as

2

A

q

AkK

q

dxdp j

jrj

j j += (2.19)

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Where j denotes the phase, K r is the relative permeability. He used a cell-to-cell non-

Darcy flow resistance flux factor, F to multiply the Darcy flow flux term, stated as

Flux non-Darcy = F ND * Flux Darcy (2.20)

n,

ND

He gave an approximate expression for the rate dependent skin factor by the expressio

jr j kK

w j r j h

D

,2

= (2.21)

numerical simulations he opined that Darcy-Forchheimer can be applied to a multiphase

system, that non-Darcy flow in occurring in the entire reservoir can be handled in a

vector form they developed the following expressions

Su35 applied his model to both oil and gas well, based on the result of his

simulator and that this model can be easily integrated with a full blown numerical

simulator.

Jamiolahmady et al. 17, when modeling flow in a crushed perforated rock, they

developed a mathematical model based on the Darcy-Forchheimer flow. From the

equation in

V V V k

P + (2

Where the gradient operator

= .22)

V is the absolute value of the velocity,

From V given aswhich they obtained an expression for

= P V

+ V k

k

1

(2.23)

The continuity equation for radial cylindrical coordinate system given as,

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( ) 01. =

+=

z V

rV r r

V z r (2.24)

Is solved to obtain an expression for V given as,

k 2

++=

k

P

V

2

411

The negative root is discarded, while the expression (2.25) is substituted in equation

(2.24). This gives

(2.25)

0

411 ++

P

k

22.

2=

P r k

(2.26)

The above expression was solved based on the finite element method using the

athematical

their model shows the limitations of the current models used in well completion

ngineering.

Femlab (COMSOL Multiphysics) m modeling software. They opined that

e

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CHAPTER III

PROBLEM STATEMENT

The productivity index of a well is a powerful tool for well evaluation. It is the

roduction rate divided by the drawdown. The productivity index, as an evaluation tool is

only valid when the well is flowing in y state (PSS) regime. Until the

pressure transient period during teady state pressure

nt

change

ity barriers or impedance (e.g scales, asphaltenes, sand

y of a

y

solution can be proffered to fix the problem. Based on the foregoing, it is obvious that a

p

a pseudo-stead

a well test is passed and a s

distribution is assumed in the well, the productivity index will not approximate a consta

with any physical significance 28.

The productivity index for an ideal well remains constant, even if the well production rate and the reservoir pressure changes during the life of the well 28. A

in the productivity index of a well over its life is an indication of an anomaly, which may

suggest the presence of permeabil

production and any other skin effect) to fluid flow in the reservoir. The productivit

well is a direct function of the pressure drop in the reservoir. Hence it is imperative to

accurately delineate and evaluate the pressure drop and know the causes of such pressure

drop in a well. This is the key goal of well performance engineering; evaluating and

calculating the pressure drop, accurately knowing the cause of the pressure drop and

designing a remedial action or proffering a solution to mitigate or remove the cause of the

pressure drop thus increasing the productivity of the well.

Therefore, in evaluating performance or non performance and in rectifying an

well problem, the source of the problem must first be identified, and then the right

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blanket description of all well problems under the Skin umbrella does not really suffice;

to adequately resolve any well problem, its source must be known. This is one of the

ain c em hallenges of this thesis; to show how poor fluid flow modeling can affect pressur

predictions and resultant effect on the calculated well productivity index.

3.1 Importance of Accurate Reservoir Pressure Prediction

The pressure profile in the reservoir is very important to reservoir and product

engineers. The production mechanism in petroleum reservoir are driven by pressure,

hence knowledge of the pressure profile is essentially an indication of the

ion

producibility ofthe reservoir. Knowledge of the reservoir pressure is important for the following reasons;

a) It gives

r reservoir properties and for

cted on wells to get one or some

the well tests.

an indication of the production mechanism of the well

b) It shows the productive capacity of the well

c) Knowing the pressure will help determine what additional equipment is

required to lift the reservoir fluid to surface.

d) It is required for reservoir management and planning.

e) Pressure profile help in determining new well locations

f) Pressure profile is a source of information fo

hydraulic connectivity.

Well tests and pressure surveys are usually condu

of the above information based on the pressure data obtained from

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3.2 Limitations of Current Techniques

A review of current industry practices as it relates to high flow rate wells was

n-

arcy flow problem in petroleum engineering still requires further research, until more

bust equations and models can be developed to solve this problem.

Although the industry actor also called

its

not applicable.

or

done in chapter 2 of this thesis. From the review it is obvious that using the historical

Darcys law to model fluid flow in high flow rate reservoir is not adequate. The no

D

ro

over the years has introduced a fudge f

the skin factor assumed to be applicable to a region of impaired permeability in the

vicinity of the well bore. This has not adequately help to narrow down the problem toroot cause and has brought in lots of uncertainties. This may explain why some remedial

jobs or work-over operations have not been successful. This is simply because the

problem was never rightly diagnosed and hence, the solution applied is

A great leap in well performance engineering will occur when well or reservoir

problems are rightly diagnosed using the right models and tools, so that the proffered

recommended solution will adequately fix the well problem at hand. The ability to rightly

calculate the individual components of the composite skin factor will help in taking

corrective measures to reduce its detrimental effect and thereby enhance the wells

productivity. Until a problem is known, it may never have a solution or it can be rightly

said that a problem known is half solved.

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3.3 Laboratory Experiments on Non-Darcy flow in Cores

The following results were obtained on core samples used in the Core Laboratory

(Corelab) of the Department of Petroleum E

ngineering Texas Tech University, to verify

e certainty of non-Darcy flows at high pressure/flow rate. The experiments were

onducted on core samples that represented different reservoir types- sandstones and

arbonates (limestone and dolomite). The experimental results for three core samples

(#13, #26 and #9) are p 3.2

Core ID: #13

th

c

c

resented in tables 3.1, 3.2 and 3.3 respectively. Figures 3.1,

and 3.3 are the graphical plot showing non-linearity in flow.

Table 3.1: Result of non-Darcy flow experiment on Core #13

Length: 6.1 cm Ambient Pressure 680.07 mmHg = 13.15 psia

Diameter: 3.745 cm Temperature 74 F

Area: 11.015 2cm Viscosity of 2 N 0.017584 cp

P ( psi ) in P (atm ) out P (atm ) Q(cc/sec) g K md Q/A L P

10 1.5765 0.8961 0.5204 7.4483 0.0472 0.1115

20 2.2569 0.8961 1.1403 8.1596 0.1035 0.2231

30 2.9373 0.8961 1.7032 8.1253 0.1546 0.3346

40 3.6177 0.8961 2.2286 7.9737 0.2023 0.4462

50 4.2981 .89610 2.8517 8.1624 0.2589 0.5577

60 4.9785 0.8961 0263 0.66923.3649 8. 0.3055

70 5.6589 3.8949 7.9631 0.78080.8961 0.3536

80 6 3.339 .8961 .11 1 0.89230 4.5368 8 6 0.4119

90 7.0197 0.89 7.9656 8 1.003961 5.0092 0.454

100 7.7001 0.8961 5.4225 7.7605 0.4923 1.1154

110 8.3805 0.8961 5.8194 7.5714 0.5283 1.2270

120 9.0609 0.8961 6.3269 7.5457 0.5744 1.3385

130 9.7413 0.8961 6.5053 7.1617 0.5906 1.4500

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Core#1 Dar

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

6 0.7

D P

3: Non- cy Plot

/ L ( a t m / c m

)

0.0 0.1 0.2 0.3 0.4 0.5 0.

Q/A (cm/s)

Figure 3.1: Experimental result of non-linearity in flow through Core #13

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Table 3.2: Result of non-Darcy flow experiment on Core #9Core ID: #9

ength: 3.55 cm Ambient Pressure 680.03 mmHg = 13.15 psiaL


Area: 10.869 Viscosity of 0.017584 cp2cm 2 N

P ( psi ) (atm ) atm ) Q(cc/sec) md Q/Ain P out P ( g K L P

10 1.5752 0.8948 0.8218 0.8097 6.984 0.0756

20 2.2556 0.8948 1.5436 0.6348 6.559 0.1420

30 2.9360 0.8948 2.3256 0.5221 6.588 0.2140

40 3.6164 0.8948 3.1546 0.4433 6.703 0.2903

50 4.2968 .89480 3.9564 0.3852 6.725 0.3640

60 4.9772 .8948 3406 0.43540 4.7323 0. 6.703

70 5.6576 5.3447 0.3052 0.49180.8948 6.489

80 6 0.338 .8948 .27 5 0.57410 6.2402 0 6 6.629

90 7.0184 0.89 0.2527 0.631548 6.8634 6.481

100 7.6988 0.8948 7.2812 0.2327 6.188 0.6699

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Core#9: Non-Darcy Plo t

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Q/A (cm/s)

D P / L ( a t m / c m

)

Figure 3.2: Experimental result of non-linearity in flow through Core #9

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Table 3.3: Result of non-Darcy flow experiment on Core #26Core ID: #26

Length: 4.145 cm Ambient Pressure 680.03 mmHg = 13.15 psia


Area: 11.04466 2cm Viscosity of 2 N 0.017584 cp

P ( psi ) in P (atm ) out P (atm ) Q(cc/sec) g K md Q/A L P

3 1.0989 0.8948 7.5020 244.19 0.6792 0.0492

4 1.1669 0.8948 9.6123 234.66 0.8703 0.0657

5 1.2350 0.8948 10.8631 212.16 0.9836 0.0821

6 1.3030 0.8948 13.0384 212.20 1.1805 0.0985

7 1.3711 0.8948 13.9540 194.66 1.2634 0.1149

8 1.4391 0.8948 15.0940 184.24 1.3666 0.1313

9 1.5071 0.8948 15.9370 172.92 1.4430 0.1477

10 1.5752 0.8948 16.9085 165.11 1.5309 0.1641

11 1.6432 0.8948 18.2907 162.37 1.6561 0.1806

12 1.7113 0.8948 19.3436 157.41 1.7514 0.1970

13 1.7793 0.8948 19.8288 148.95 1.7953 0.2134

14 1.8473 0.8948 20.9396 146.06 1.8959 0.2298

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Core#26: Non-Darcy Plot

0.00

0.05

0.10

0.15

0.20

0.25

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Q/A (cm/s)

D P / L ( a t m / c m

)

Figure 3.3: Plot of Experimental result of non-linearity in flow through Core #26

3.4 Problem Statement

The buildup of the thesis up till now as been to lay the foundation of flow in

porous media, describe the peculiarities of Darcy and non-Darcy flows, review current

industry practice and show there inadequacies. This has been a gradual crescendo to the

petroleum engineering problems this thesis seeks to investigate and proffer a solution to;

these problems are summarized in the following statements. The inadequacy of Darcys

law to model fluid flow in reservoirs with high velocity flow profiles and the resultant

error it propagates in well performance analysis.

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The traditional use of the rate-dependent skin factor to account for the additional

pressure loss due to high velocity flows, neglects pressure losses in the reservoir, since it

only assumes that the losses are important in the vicinity of the well bore, research has

shown that this is not the case especially in fractured reservoirs.

There is no proven method of knowing flow regimes in the reservoir; thus

obfuscating the judgment of a well analyst in flow modeling.

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CHAPTER IV

SOLUTION STATEMENT

4.1 Proposed Solution

The previous chapters has adequately shown the importance and gravity of the

non-Darcy flow phenomena, and highlighted the scenario where this phenomenon occurs

in the prospect of oil and gas. The obvious limitations of the Darcys law as a flow

modeling equation for these scenario is evident.

The proposed solution is to integrate the Darcy-Forchheimer equation into the

flow modeling equation for non-linear (high velocity flows), and use the developedequation to model fluid flow in the reservoir, especially for non-linear flows. The

productivity index of the well is then calculated using this model, with the objective that

a more representative well productivity will be obtained in these scenarios.

4.2 Derivation of the Mathematical Model

In chapter 1, the three fundamental equations required to model fluid flow in any

media were stated as:

a) Continuity equation (Law of conservation of mass)

b) Equation of state

c) Equation of motion/dynamics (Flow Equation)

The derivation of the non-linear mathematical flow equation is given below:

The continuity equation, assuming constant porosity is given by,

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0)( =+

vdivt

r

(4.1)

)( vdivt

r

=

vvdivt

rr 1)(

=

(4.2)

From product rule,

x P

P x

t P

P t

=

=

Substituting these expressions in equation (4.2) above,

,

)(

g Simplifyin

v x P

P vdiv

t P

P rr

=

P vvdivt

P =

rr 11 )( (4.3)

Equation (3) above is the final form of the continuity equation used.

The equation of flow is the Darcy-Forchheimer equation given by:

2vvk dx

dp +=

And in vector form as, letk

= , then the expression becomes

P vvv =+ rrr

0=++ vvv P rrr

(4.4)

The equation of state is given by the expression;

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(4.5) 1' =

Where ( is the compressibility))(0 01 P P e

= 1

Equations (3), (4) and (5) are the three governing equations to be used in the derivation of

the mathematical framework for the model.

The vector velocity ),( t xvr

cannot be uniquely represented as a function of the pressure

gradient , we assume an approximation given by; P

P P f vvvvv === )(),,( 321 rr

Correspondingly,

P P f v = )(

Substituting these in the Darcy-Forchheimer equation, equation (4) above,

0)))(()((1(

0)(.)())((

2 =++

=++

P P f P f P

P P f P P f P P f P

This is a form of a quadratic equation, therefore solving for )( P f , and taking only the

positive root as the valid solution to the equation, this results in

P

P P f

++=

2

4)(

2

Multiplying the numerator and the denominator by ( ) P ++ 42 , results in

P P f

++=

42)(

2 (4.6)

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Equation (6) above is a solution of the velocity vector vr

of the Darcy-Forchheimer

equation.

The continuity equation for slightly compressible fluid from equation (4.3), is given by

)(' vdivt

P r =

(4.7)

For slightly compressible fluids, the term ( P v r1 ) is negligible,

Substituting Darcy-Forchheimer parameters into equation (4.7), results in

))(( P P f divt

P =

(4.8)

This is the form of the partial differential equation (PDE) that is used to model the non-

linear Darcy-Forchheimer flow in porous media.

In developing this model, the following assumptions have been made:

a. Pressure independent rock and fluid properties

b. Homogenous and isotropic porous medium with uniform thickness

c. Negligible gravity forces

4.3 Description of the Simulator

The software used in solving the PDE above is called COMSOL Multiphysics. It

a commercial package used in solving systems of partial differential equations (PDE),

typically seen in scientific and engineering problems. The solution of the PDE is based

on the finite element method (FEM) scheme for solving PDEs. The software runs the

finite element analysis with adaptive meshing and error control using a variety of

numerical solvers.

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In COMSOL Multiphysics, PDEs can be described in three ways;

a) Coefficient form: Suitable for linear or nearly linear models

b) General form: Suitable for nonlinear models

c) Weak form: For PDEs on boundaries, edges or points or for models with

mixed space and time derivatives.

The coefficient form of PDE model was used for solving the Darcy-Forchheimer

nonlinear model, in this thesis.

4.4 Numerical Computation and AlgorithmThe Darcy-Forchheimer model was applied to different reservoir geometry to

evaluate the productivity indexes of these reservoirs. A comparison is made between the

cases when Darcys law is used versus when the Darcy-Forchheimer model was used to

model flow in the reservoir. The reservoir geometry used were obtained from reservoir

geometries for which shape factors have been obtained for pseudo-steady state

productivity index calculation as stated in chapter. The flow chart in figure 4.1 is a

diagrammatic representation of the steps used in solving the model, using COMSOL

Multiphysics.

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49

Define form of PDE

Draw Reservoir andWell Geometry

Define BoundaryConditions and InitialValues of Parameters

Input Solve ParametersSelect Solver Type

Generate Plot of OutputData in EXCEL

Enter ModelingEquation and

Reservoir Domain

Input Values of Constantsand Parameters

Read off Output DataPressure

Productivity Index (PI)

Is Output:same?

End ofRoutine

Define Grid Size(Initialize or Refine Grid

Mesh Size)

NO

YES

Figure 4.1: Flow Chart of Numerical Computation

COMSOLMultiphysics

Initialize


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4.5 Laboratory Measurement of Beta Factor

The Laboratory measurement of the Beta factor was done by first measuring the

absolute permeability of the core samples used in the experiments then increasing the

pressure drop across the cores at an ever increasing pressure differential while measuring

the flow rate. The experimental set up is shown diagrammatically in figure 4.2 below.

A linear version of the Forchheimer equation was then used to calculate the

coefficient of inertial resistance, beta. (This procedure is described by Dake 8 in his book,

Fundamentals of Reservoir Engineering, page 259).

Figure 4.2: Experimental setup for permeability and factor measurements

The experimental procedure used is presented diagrammatically flow in figure 4.3 below.

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51

Start

Measure porosity of core samplesusing helium porosimeter

Prepare core samples formeasurement

Sort cores intogroups according

to porosities

Use Klinkenberg correction to obtainabsolute permeability (K L)

Measure gas permeability (K g) usingnitrogen gas at low pressures (flow rate)

Are porosities insame range?

+= v

k dxdP

v

2

1

Apply increasing pressure differentials across coresample and record flow rate

Obtain beta factor fromDarcy-Forchheimer equation

End

Plot beta as a function ofabsolute K on a Log-Log graph

Express beta as a function ofabsolute permeability K

No

Yes

k C =

Figure 4.3: Procedure for Laboratory Measurement of Factor


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The absolute permeability of the cores was obtained by first measuring gas

permeability using nitrogen gas, and then applying the Klinkenberg correction to obtain

the absolute permeability of the core samples.

Initially 26 core samples were sampled for the experiments, but after measuring

the core porosities, it was decided to carry out permeability measurement only on ten

core samples sorted based on their porosities and initial permeability tests. Table 4.1

below is the spreadsheet used for the porosity calculations. Porosity was measured using

the Helium porosimeter.

Table 4.1: Porosity, physical properties and Lithology of core samples usedLithology

Core ID#

Diameter(cm)

Length(cm)

Bulk Volume(cc) Porosity

Sandstone 1 3.720 3.4650 37.660 0.1829Sandstone 2 3.720 3.6500 39.671 0.0909Sandstone 3 3.700 3.6100 38.815 0.1730Sandstone 4 3.740 3.9650 43.559 0.1420Sandstone 5 3.720 3.4400 37.388 0.1699Sandstone 6 3.720 3.3000 35.867 0.1812Sandstone 7 3.720 3.4000 36.953 0.1247Sandstone 8 3.720 3.9450 42.877 0.1246Sandstone 9 3.720 3.5500 38.584 0.1838

Sandstone 10 3.725 3.2800 35.745 0.1850Sandstone 11 3.700 5.0800 54.621 0.1017Sandstone 12 3.700 5.5950 60.158 0.0756Sandstone 13 3.745 6.1000 67.193 0.1377Sandstone 14 3.740 5.1500 56.577 0.1323Sandstone 15 3.745 3.9400 43.400 0.1030Sandstone 16 3.745 5.6400 62.126 0.1050Sandstone 17 3.745 6.2700 69.065 0.0812Carbonate 18 3.755 6.2000 68.660 0.0629Carbonate 19 3.740 5.1000 56.028 0.1402Carbonate 20 3.745 3.2300 35.579 0.0166Carbonate 21 3.800 5.7700 65.438 0.1114Carbonate 22 3.750 4.9400 54.561 0.1340Carbonate 23 3.780 5.4400 61.048 0.1368Carbonate 24 3.750 5.0000 55.223 0.0819Carbonate 25 3.770 4.4250 49.395 0.1457Carbonate 26 3.750 4.1450 45.780 0.0992

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The core samples were ranked based on their porosities and an initial permeability

measurement done on the core samples to select the cores that were used in the final

analysis. The core selection is given the table 4.2 below.

Table 4.2: Porosity ranking and cores used for permeability measurementsLitholo gy Core # Porosity CommentsSandstone 10 0.1850Sandstone 9 0.1838Sandstone 1 0.1829Sandstone 6 0.1812Sandstone 3 0.1730Sandstone 5 0.1699Carbonate 25 0.1457Sandstone 4 0.1420Carbonate 19 0.1402Sandstone 13 0.1377Carbonate 23 0.1368Carbonate 22 0.1340Sandstone 14 0.1323Sandstone 7 0.1247Sandstone 8 0.1246Carbonate 21 0.1114Sandstone 16 0.1050Sandstone 15 0.1030Sandstone 11 0.1017Carbonate 26 0.0992 Highly Fractured

Sandstone 2 0.0909Carbonate 24 0.0819Sandstone 17 0.0812Sandstone 12 0.0756Carbonate 18 0.0629Carbonate 20 0.0166 Fractured

Core #26 was selected because it is highly fractured and it will serve as a good candidate

to investigate non-Darcy flow in fractured reservoir.

The absolute permeability of the core samples is given in table 4.3 below; the results and

analysis of the laboratory measurements are given in appendix A.

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Table 4.3: Porosity and Permeability of Core samples used in factor experimentCore ID Porosi ty Permeabil ity (md)

10 0.1850 5.3625

9 0.1838 6.1820

1 0.1829 5.04866 0.1812 1.7786

3 0.1730 3.8944

25 0.1457 2.1851

13 0.1377 7.5883

23 0.1368 3.2689

22 0.1340 0.8449

26 0.0992 160.39

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CHAPTER V

RESULTS OF NUMERICAL COMPUTATIONS

The results of numerical computations using COMSOL Multiphysics, is presented

in this chapter. Different reservoir geometry and well configurations were used in the

computations. The dimensions of the reservoir and the well are given for each of the

geometry used in the computation.

5.1 Horizontal Well in a Rectangular Reservoir

The first geometry used in the numerical computation is a horizontal drain-hole ina rectangular reservoir. Figure 5.1 shows the location of the horizontal drain-hole relative

to the boundaries of the reservoir, as shown it is located in the center of the reservoir. The

dimensions used for the computation are stated below.

Dimensions: Length = 800 meters

Width = 400 meters

Well radius = 15 cm (6 inches)

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Figure 5.1: Geometry of the Horizontal Drain in a Rectangular reservoir (Geometry 5.1)

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The results of the numerical computations of geometry 5.1 are given in table 5.1.

It is the result of the variation of the calculated productivity index of the reservoir

geometry as length of the horizontal drain-hole and factor are varied for the geometry.

Table 5.1: Productivity Index at different drain-hol

amao abiodun thesis

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